Question

A platter is in the shape of a regular octagon. Find the perimeter and area of the platter if the apothem is 6 inches.

Answer 1

Perimeter of platter is 39.84 inch.

Area of platter is
`119.52\:inch^(2)`

Explanation:

Apothem is the perpendicular line from center to the side of the octagon. Given that apothem is 6 inch that is AB=6 inch and
`m\angle 90^(\circ)` (Refer attachment).

Assume DB = BC = x.

Since
`\angle DAC` is central angle, Therefore

`m\angle DAC=(360)/(8)`

`m\angle DAC=45^(\circ)`

Since apothem AB bisects angle DAC. So,

`\angle BAC=( m\angle DAC )/(2)`

`\angle BAC=( 45 )/(2)`

`\angle BAC=22.5^(\circ)`

To find the value of BC, use trigonometry ratio for tan for
`\Delta ABC`,

`tan\:\theta=(opposite\:side)/(adjacent\:side)`

`tan\left(22.5\right^(\circ))=(BC)/(AB)`

`tan\left(22.5\right^(\circ))=(x)/(6)`

Solving for x multiply whole equation by 6.

`tan\left(22.5\right^(\circ))=(x)/(6)`

`6* tan\left(22.5\right^(\circ))=x`

`x=2.49`

Now, DC = DB + BC

`DC = x + x`

`DC = 2 x`

`DC = 4.98`

Therefore, side of octagon is 4.98 inch .

Perimeter of octagon is,

`Perimeter=number\:of\:sides* length`

Since octagon has 8 sides, so number of side is 8

`Perimeter=8* 4.98`

`Perimeter=39.84`

Therefore, perimeter of platter is 39.84 inch.

Area of octagon is,

`Area\:of\:octagon=(1)/(2)* apothem* perimeter`

`Area\:of\:octagon=(1)/(2)* 6* 39.84`

`Area\:of\:octagon=119.52`

Therefore, area of platter is
`119.52\:inch^(2)`

Note: Values are calculated using two decimal places)